Home page: www.math.u-szeged.hu/phd/

**Chair:** ** ****Prof. Gábor Czédli DSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 093, fax: (+36 629 544 548,

E-mail: czedli@math.u-szeged.hu

WWW: http://www.math.u-szeged.hu/~czedli

**Assistant chair: Assoc. Prof. Péter Hajnal CSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 088, fax: (+36 62) 544 548,

E-mail: hajnal@math.u-szeged.hu

WWW: http://www.math.u-szeged.hu/~hajnal

**Assoc. Prof. László Zádori DSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 543 881, fax: (+36 62) 544 548,

E-mail: zadori@math.u-szeged.hu

WWW: http://www.math.u-szeged.hu/~zadori

**Application procedure**

- Application requirements: at least intermediate level (B2) certified by an internationally accredited language examination (TELC/ECL B2, TOEFL IBT 72, RPDT 72, Cambridge FCB, IELTS 5.5)
- Application deadline: 15 May
- Application fee: 35 EUR

- Tution fee: max. 5500 EUR/semester
- Entrance examination fee: 100 EUR

**For further information please contact**:

Faculty of Science, Bolyai Institute

Phone: +36-62-544-093, fax: +36-62-544-548

czedli@math.u-szeged.hu

Website: www.math.u-szeged.hu/phd/

Mathematical education and research at the Bolyai (Mathematical) Institute of the University goes back to Frigyes Riesz and Alfréd Haar, 1921. These two world famous masters of mathematical analysis founded “Acta Scientiarum Mathematicarum (Szeged)” in 1922, the first mathematical journal in foreign language in Hungary. The “Acta” played and continues to play a crucial role in the circumstance that the Library of Bolyai Institute offers more than 250 current mathematical journals, almost 20 thousand books and more than 20 thousand volumes of journals to professors, researchers and students. Today the Bolyai Institute consists of six chairs: Algebra and Number Theory, Analysis, Applied and Numerical Mathematics, Geometry, Set Theory and Mathematical Logic, Stochastics. The Institute has 50 academic staff members with a wide range of research interests.

The doctoral education in the Bolyai Institute started in 1993. Between 2002 and 2008, the School consisted of two educational programmes: mathematics and informatics. The school reached its present form in 2009.

__Exemption from English language exam certificate:__

During the application to the programmes of the Faculty of Science and Informatics those applicants are not required to provide the English language exam certificates who are natives of one of the following countries and can officially demonstrate that their academic studies have been conducted in English.

However, these students will also have to obtain an international language proof B2 level by the end of their studies or for the PhD process.

__Countries concerned:__

- Antigua and Barbuda
- Anguilla
- Australia (also Christmas Island, Norfolk Island, Keeling Islands, Cocos Islands)
- Bahamas
- Barbados
- Belize
- Bermuda
- Botswana
- British Antarctic Territory (also British Antarctica, Antarctica British)
- British Indian Ocean Territory (also Chagos Archipelago, Indian Ocean Territory (British))
- British Virgin Islands
- Canada
- Cayman Islands
- Channel Islands
- Dominica
- Falkland Islands
- Ghana
- Gibraltar
- Grenada
- Guyana
- Irish Republic
- Jamaica
- Kenya
- Lesotho
- Malta
- Montserrat
- Namibia
- New Zealand (also Niue, Tokelau, Cokelau, Cook Islands)
- Nigeria
- Seychelles
- Singapore
- St Kitts and St Nevis
- St Lucia
- St Vincent and the Grenadines
- Trinidad and Tobago
- Turks and Caicos Islands
- Uganda
- UK
- USA
- Zimbabwe

**Educational Programmes**

Each of the five PhD educational programmes, listed below, takes minimum 3 years, the average duration is 4-5 years. The students have to choose at least 8 courses such that at least 3 of them are general or basic courses. The Bolyai Institute offers 5 general courses (*Algebra*,* **Theory of Measure and Integral*, *Topology,* *Discrete Mathematics, Probability Theory*), 27 basic courses and several dozens of specialized courses. Two basic courses and 11 specialized courses are devoted to the Didactics of Mathematics. The lectures are taught mostly by the members of the Bolyai Institute or, sometimes, by invited experts. If the enrolment is low, a course may be held as a reading course with regular consultations, possibly in English.

**1.** ** ****Algebra**

**Programme director:**** ****Assoc. Prof. László Zádori DSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary Phone: (+36 62) 543 881, fax: (+36 629 544 548,

E-mail: zadori@math.u-szeged.hu

http://www.math.u-szeged.hu/~zadori

– Semigroup theory: regular semigroups and their classes and generalizations, inverse semigroups, the structure of simple semigroups.

– Lattice theory, congruence lattices and Mal’cev conditions, related lattices, lattices and their generalizations.

– Universal algebra, clones and relations, structure theory of finite algebras, varieties, commutator theory, finitely and nonfinitely based algebras and varieties, algebraic methods in algorithmic problems.

– The antique and medieval roots of the secondary (high) school curriculum in mathematics.

Supervisors: B. Csákány DSc, G. Czédli DSc, L. Klukovits CSc, M. Maróti PhD, L. Megyesi CSc, L. Szabó CSc, Á. Szendrei DSc, M. B. Szendrei DSc, L. Zádori DSc.

MDPT211. Semigroup theory (Mária B. Szendrei, DSc)

Literature: Grillet, P. A.: Semigroups: An Introduction to the Structure Theory, Marcel Dekker, 1995. , Howie, J. M.: Fundamentals of Semigroup Theory, Clarendon Press, 1995.

MDPT212. Lattice theory (Gábor Czédli, DSc)

Literature: G. Czédli G.: Lattice Theory (Hungarian), G. Grätzer: General Lattice Theory

MDPT213. Universal algebra (László Zádori, DSc)

Literature: Burris–Sankappanavar: Introduction to Universal Algebra, McKenzie–McNulty–Taylor: Algebras, Lattices, Varieties.

MDPT214. Group theory (Ágnes Szendrei, DSc)

Literature: Aschbacher: Finite Group Theory. Hall, M. Jr.:The Theory of Groups. Huppert: Endliche Gruppen. Lyndon-Schupp: Combinatorial Group Theory.

**2.** ** ****Analysis**

**Programme director:**** ****Prof. Ferenc Móricz DSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 082, fax: (+36 62) 544 548,

E-mail: moricz@math.u-szeged.hu

**Research topics**

– Approximation theory and orthogonal polynomials

– Fourier series, integrals and orthogonal series

– Functional analysis

– Operator theory

– Summability and Tauberian theorems

– Some interesting problems of the analysis and their treatment during teaching

Supervisors: F. Móricz DSc, L. Kérchy DSc, L. Leindler MHAS, J. Németh CSc, L. Stachó CSc, V. Totik MHAS.

**Basic courses**

MDPT221. Chapters from the Complex Function Theory (László Leindler, MHAS)

*Mittag-Leffler’s theorem on the decomposition of meromorphic functions into partial fractions, decomposition of** *cot*πz. Weierstrass factorization theorem of entire functions, factorization of** *sin *πz. The gamma function. Approximation by rational functions, Runge’s theorem. Hardy Hp spaces of analytic functions on the open unit disk. Nontangential limit on the unit circle, Fatou’s theorem. Riesz brothers’ theorem, Szegő’s theorem. Blaschke products, inner and outer functions, canonical factorization. Banach algebra of the functions that are analytic on the open unit disk and continuous on the closed unit disk. Invariant subspaces of the shift operator in the Hilbert space H2.** *

Compulsary literature: B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions. K. Hoffmann: Banach spaces of analytic functions. Recommended literature: P. L. Duren: Theory of *Hp* spaces, J. Garnett: Bounded analytic functions. P. Koosis: Introduction to *Hp* spaces, W. Rudin: Real and complex analysis.

MDPT223. Introduction to the Approximation Theory (Vilmos Totik, MHAS)

*Approximation by positive operators, Korovkin’s theorem, the Weierstrass and Weierstrass-Stone theorems. Moduli of continuity and smoothness, Jackson’s theorem, direct theorems. Estimation of the derivatives, Bernstein’s theorem and the inverse theorems of approximation theory. Characterization of the best approximating polynomials, extreme signatures. Lp-approximation. Saturation of Bernstein polynomials, parabola method.** *

Compulsary literature: G. G. Lorentz: Approximation of functions, G. G. Lorentz–R. DeVore: Approximation theory, Recommended literature: N. I. Akhiezer: Lectures on the theory of approximation, P. Borwein–T. Erdelyi: Polynomials and Polynomial Inequalities, G. G. Lorentz–M. Von Golitschek-A. Makovez: Approximation Theory II., I.P. Natanson: Constructive function theory, M. Timan: The approximation of real functions.

MDPT224. Fourier Series I. (Zoltán Németh, PhD)

*Completeness of the trigonometric system. Bessel’s inequality, Parseval’s formula. Convergence of Fourier series: Riemann-Lebesgue lemma, the Dini test, the principle of localization, the Dirichlet-Jordan test, the Lebesgue constants. Summability of Fourier series: theorem of Fejér and its consequences, theorem of Lebesgue, the Lebesgue points of an integrable function. Divergence of Fourier series: examples of Fejér and Lebesgue. Special trigonometric series with coefficients tending monotonically to zero.** *

Compulsary literature: A. Zygmund: Trigonometric series, Recommended literature: N. K. Bary: A treatise on trigonometric series, R. E. Edwards: Fourier series, B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions.

MDPT225. Functional Analysis (László Kérchy, DSc)

*Orthonormal systems in Hilbert spaces, dimension of a Hilbert space. Convergence of Fourier series, Cesàro and Abel summability. The Hahn-Banach theorem and its applications: the Banach limit, integral and measure. The Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem: their applications to Fourier series. Duals of the Lp spaces, reflexivity. The dual of the space of continuous functions, the Riesz representation theorem. The Weierstrass-Stone approximation theorem.*

Compulsary literature: L.Kérchy: Real and Functional Analysis, Polygon, Szeged, 2008 (Hungarian), Recommended literature: F. Riesz–B. Szőkefalvi-Nagy: Functional analysis, W. Rudin: Real and complex analysis, B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions.

**Three representatives from the specialized courses**

MDPT3203. Contractions of Hilbert spaces II (László Kérchy, DSc)

*Operator-valued analytic functions. Inner and outer functions, factorization theorems. Scalar multiple. The characteristic function and the function model of a contraction. The connection between spectra and characteristic functions. Characterizing invariant subspaces of contractions with regular factorizations of characteristic functions. Invariant subspaces of C11-contractions. Weak contractions.*

Compulsary literature: B. Szőkefalvi-Nagy–C. Foias: Harmonic analysis of operators on Hilbert spaces, Recommended literature: H. Bercovici: Operator theory and arithmetic in *H*infinity. C. Foias–A.F. Frazho: The commutant lifting approach to interpolation problems, K. Hoffmann: Banach spaces of analytic functions.

MDPT3214. Complex harmonic analysis I (Ferenc Móricz, DSc)

*Representation of holomorphic functions on the open unit disk by Poisson integrals. Holomorphic completion of harmonic functions, Herglotz integral. Hp and hp spaces on the unit disk of the complex plane. Characterization of the h*1 *space by Poisson-Stieltjes integral. The existence of the limit functions of a function in h*1* **. Holomorphic definition of the logarithms of a holomorphic function. The Jensen and Poisson-Jennsen formulas, distribution of the zeros of an analytic function. The existence and characterization of Blaschke products, F. Riesz’ and Nevalinna’s factorization theorems. Factorization of an inner function. The existence of the limit functions of a function in N. Convergence to the limit function in Lp-norm. Characterization of Hp by Poisson integal, the Riesz brothers’theorem and its equivalent reforumulations. The existence of the outer function. Canonical factorization of a function in Hp.*

Compulsary literature: P. L. Duren: Theory of *Hp* spaces, Recommended literature: Garnett: Bounded analytic functions, K. Hoffmann: Banach spaces of analytic functions, P. Koosis: Introduction to *Hp* spaces, A. Torchinsky: Real variable methods in harmonic analysis, A. Zygmund: Trigonometric series

MDPT3225. Approximation by polynomials (Vilmos Totik, MHAS)

*Trigonometric polynomials, Nikolskii’s theorems, Dzjadik’s inverse theorems, characterization of the best approximation by algebraic polynomials in terms of φ-modulus, discrete operators, potential theory and approximation by polynomials; approximation by changing weights, orthogonal polynomials and weighted approximation by polynomials; Müntz’ theorem and its generalizations.** *

Compulsary literature: Z. Ditzian–V. Totik: Moduli of smoothness, G. G. orentz–R. DeVore: Approximation theory, M. Timan: The approximation of real functions, Recommended literature: N. I. Akhiezer: Lectures on the theory of approximation, G. G. Lorentz: Approximation of functions, I.P. Natanson: Constructive function theory

**3.** ** ****Dynamical**** ****Systems**** **

**Programme director:**** ****Prof. László Hatvani MHAS**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 079, fax: (+36 62) 544 548,

E-mail: hatvani@math.u-szeged.hu

http://www.math.u-szeged.hu/~hatvani

**Research topics**

– Qualitative theory of ordinary, functional, and partial differential equations: nonlinear oscillations; existence and stability of periodic solutions; almost periodic solutions

– Lyapunov stability and its applications; stability in population dynamics; dependence of the solutions from parameters for partial differential equations; radially symmetric solutions of elliptic partial differential equations

– Existence and smoothness of invariant manifolds for functional differential equations; stable subharmonic solutions of periodic functional differential equations; existence of heteroclinic orbits; infinite dimensional dissipative dynamical systems; Hopf bifurcation and centre manifolds; inertial manifolds; exponential attractors

– Numerical solutions; computer simulation of dynamical systems

– Modern didactical methods in the teaching of dynamical systems

Supervisors: L. Hatvani MHAS, T. Csendes DSc, J. Karsai PhD, T. Krisztin DSc, B.G. Pusztai PhD, G. Röst PhD.

**Basic courses**

MDPT231–232. Ordinary differential eqations I-II (László Hatvani, MHAS)

*Differential equations on manifolds. Existence and uniqueness theorems. Differential equations in spaces of infinite dimension. Linear systems. Infinitesimal generator. Integral manifolds. Linearization, Hartman-Grobman’s theorem. Perturbation theory. Non-autonomous systems. Periodic and almost periodic equations. The method of averaging. Boundary value problems. Sturm-Liouville theory. Second order equations, oscillation. Limit sets and limit cycles. Poincaré-Bendixson theorem. Stability. Lyapunov’s methods. Invariance principles. First order partial differential equations. Hamilton-Jacobi’s theory.** *

Compulsary literature: H. Amann, Ordinary Differential Equations, DeGruyter, 1990. V. I. Arnold, Ordinary Differential Equations, Springer-Verlag, 1992. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969. Recommended literature: D. V. Anosov, V. I. Arnold, Dynamical Systems I, Ordinary Differential Equations and Smooth Dynamical Systems, Springer-Verlag, 1991. C. Chicone, Ordinary Differential Equations with Applications, Springer, 1999. Ph. Hartman, Ordinary Differential Equations, Birkhäuser, 1982. M. Hirsch, S. Smale, Differential Equations, Dynamic Systems and Linear Algebra, Academic Press, 1974. M. A. Naimark, Linear Differential operators, Nauka, 1969 (in Russian). V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, 1954. J. Palis, W. DeMelo, Geometric Theory of Dynamical Systems, Springer-Verlag, 1982. V. A. Pliss, Integral Manifolds of Periodic Systems of Differential Equations, Nauka, 1977 (in Russian).

MDPT233. Partial differential equations I (Tibor Krisztin, DSc)

*Distributions (generalized functions). Sobolev spaces. Fourier transform of distributions. Fundamental solutions of PDE’s. Partial differential operators. Classical and generalized solutions. Hypo-elliptic differential operators. Well-posed problems in half space for linear systems of PDE’s. Existence, uniqueness and stability of solutions of boundary value problems for elliptic, hyperbolic, and parabolic PDE’s in Sobolev spaces.** *

Compulsary literature: V. Sz. Vlad’imirov, B Introduction to the theory of partial differential equations. L. C. Evans, Partial Differential Equations, Doctoral Studies in Mathematics, Vol. 20, AMS, Providence, Rhode Island, 1998. Recommended literature: O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, 1985. I. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1986.

MDPT234–235. Dynamical systems I-II (László Hatvani, MHAS)

*Existence and smoothness of invariant manifolds. Behaviour of solutions near fix points and periodic orbits. Linearization. Orbital stability. Poincaré maps. Averaging. Limit sets. Asymptotically smooth maps and semi-groups. α-contractive semi-groups. Stability of invariant sets. Dissipative systems. Global attractors. Fix point theorems. Morse-Smale maps. Dimension of a global attractor. Periodic flows. Gradient systems. Examples: retarded FDE’s, neutral FDE’s, parabolic and hyperbolic PDE’s.** *

Compulsary literature: M. Hirsch and S. Smale, Differential Equations, Dynamic Systems and Linear Algebra, Academic Press, 1974. J. Palis, W. DeMelo, Geometric Theory of Dynamical Systems: an Introduction, Springer-Verlag, 1982. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990. Recommended literature: J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, 1983. J. Hale, L. Magalhaes, W. Oliva, An Introduction to Infinite Dimensional Dynamical Systems – Geometric Theory, Springer-Verlag, 1984. J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1986. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981. M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds, Springer-Verlag, 1977. V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, 1954. H. L. Smith, Monotone Dynamical Systems, AMS, 1995. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997.

**A representative from the specialized courses**

Functional differential equations I, II (Tibor Krisztin, DSc)

*The abstract theory of phase spaces, trajectories, solutions. Existence and uniqueness theorems. Continuous dependence on the initial data. New phenomena in comparison with the ordinary differential equations. Continuability and compactness of solutions. Linear functional differential equations. Oscillation for first and second order equations. Stability. Integro-differential equations. Neutral equations. Geometric theory of autonomous equations. The existence of periodic solutions. Applications from biology, mechanics, and other sciences.** *

Compulsary literature: O. Dickmann, S. A. Van Gils, S. M. Verduyn Lunel, H.-O. Walter, Delay Equations, Springer, 1995. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977., Recommended literature: T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, 1985., G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990. I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, 1991., Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, 1991. V. B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986., T. Krisztin, H.-O. Walter, J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, AMS, 1999., S. H. Saperstone, Semidynamical Systems in Infinite Dimensional Spaces, Springer-Verlag, 1981.

**4. Geometry, Combinatorics and Theoretical Computer Science**

**Programme director:**** ****Assoc. Prof. Péter Hajnal CSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1. Hungary

Phone: (+36 62) 544 088, fax: (+36 62) 544 548,

E-mail: hajnal@math.u-szeged.hu

http://www.math.u-szeged.hu/~hajnal

**Research topics**

– Differential geometry, Lie groups and Lie algebras; analysis on manifolds; web geometries.

– Discrete geometry and combinatorics; finite geometry; combinatorial and convex geometry; latin squares and 3-webs.

– Graph theory; combinatorial complexity theory.

– Elementary geometry and combinatorics.

– Theoretical computer science

Supervisors: P. Hajnal CSc, J. Barát PhD, T. Csendes DSc, J. Csirik DSc, J. Dombi CSc, Z. Ésik DSc, F. Fodor PhD, Z. Fülöp DSc, G. Gévay PhD, Cs. Imreh PhD, J. Kincses CSc, Gy. Kiss PhD, G. P. Nagy PhD, A. Pluhár PhD, Gy. Turán CSc.

**Basic courses**

MDPT241.Combinatorial methods in geometry (György Kiss, PhD)

*Block designs. Parameters of block designs and divisibility conditions. Steiner systems. Hadamard matrices. Solvable block designs. Baranyai’s theorem. Finite projective geometries. Latin squares. Combinatorial properties of finite projective geometries. Coordinatization of finite planes, the theorems of Desargues and Pappus. Finite affine planes. Finite reflection groups. Coxeter groups and complexes. Buildings.** *

Recommended literature: M. Jr. Hall, Combinatorial theory, Waltham, Mass. 1967., Gy. Kiss and T. Szőnyi: Finite Geometries, Polygon, Szeged, 2001 (Hungarian), P. Hajnal: Systems of sets, Polygon, Szeged, 2002 (Hungarian), Brown, Buildings, Springer-Verlag, London, 1989.

MDPT242. Riemannian geometry (Árpád Kurusa, CSc)

*Riemannian geometry: metric, Levi-Civita connection, curvature. Geodesics: normal** **neighborhood*, *convex neighborhood*, *normal coordinates*, *geodesic variation.** **Jacobi fields, the Jacobi theorem on the characterization of minimizing geodesics via conjugate points. Hopf-Rinow theorem. Hadamard theorem.** **Morse index theorem. curvature tensor*. *Ricci curvature*, *scalar curvature*, *sectional curvature. Spaces of constant** **curvature.*

Compulsary literature: M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992., J. Milnor, Morse Theory, Princeton University Press, 1963., Recommended literature: W. Klingenberg, D. Gromoll, W. Meyer: Riemannsche Geometrie im Grossen, Springer, 1968., J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, 1975.

MDPT243. Convex bodies and classical integral geometry

*Basic properties of convex sets. Charatheodory, Radon and Helly’s theorem. Separability, Euler’s relation, duality. Approximation of convex sets, Blaschke’s choice theorem. Mixed volume, Brünn-Minkowski theorem,** **Fenchel-Alexandrov inequality.** **Density for points and lines, cinematic density, planar integral formulas, Steiner’s formula, quermass integrals. Blaschke and Poincaré’s fundamental formulas. Curve integrals and their applications.** *

Compulsary literature: L.A. Santalo, Integral Geometry and Geometric Probability, Encyclopedia of Math., Addison–Wesley, London, 1976. T. Bonnesen, W.Fenchel, Theorie der konvexen Körper, Springer, Berlin, 1934. , W. Blaschke, Vorlesungen über Integralgeomtrie, Berlin, 1955. H. Busemann, Convex surfaces, Interscience, London, 1958.

MDPT244. Computational geometry (Péter Hajnal, CSc, and Ferenc Fodor, PhD)

*Special data structures for geometric problems. Geometric search. Coding polytopes, arrangements. Allowable sequences. Range search, incremental algorithms. Complexity of cell structures. Convex hull algorithms in two and three dimensions. Average case analysis of algorithms. Geometry of linear programming. Point location in arrangements. Maximal inscribed sphere, minimal volume enclosing simplex. Maximization of vector sums. Similarity search. Algorithms for determining Voronoi diagram, Delaunay triangulation Triangulations. Closest neighbor problem Determining the minimal length spanning tree. Shape of point sets. Separation and intersection problems. Design of algorithms.** *

Recommended literature: M. de Berg, M. van Kreveld, M. Overmas, O. Schwarzkopf: Computational Geometry, 2nd. revised edition, Springer 2000., H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, New York, 1987.

MDPT245. Geometric algebra (Gábor P. Nagy, PhD)

*Affine and projective spaces. Desargue theorem and coordinatizing with fields. Pappos theorem and commutativity of the coordinatizing field.The characteristic of the coordinatizing field and the Fano configuration.Collinearity*-*preserving functions, semilinear maps. Sympletic and othogonal geometry. The structure of the symplectic and orthogonal groups. Clifford algebra.*

Compulsary literature: E. Artin, Geometric Algebra, Princeton University, 1957., R. Baer, Linear Algebra and Projective Geometry, Academic Press, 1952., Recommended literature: D. R. Hughes, F. C. Piper: Projective Planes, Springer, 1970., J. Dieudonné, La Géométrie des Groupes Classiques, Springer, 1955., Gy. Kiss and T. Szőnyi: Finite Geometries, Polygon, Szeged, 2001. (Hungarian)

MDPT246. Algebraic topology (János Kincses, CSc)

*Simplicial complexes and homotopy. Baricentric division and simplicial approximation theorem. The fundamental group and computational methods for it. 2-dimensional triangulable manifolds and their characterization. Singular homology groups and algorithmic methods for them: simplicial homology, exact sequences. Homology with arbitrary coefficients. Lefschetz fixed point theorem. Cohomology groups and their computation. Alexander-Poincare duality. Homotopy theory of CW-complexes Whitehead theorem and cellular approximation. Homology and cohomology of CW-complexes.Hurewitz theorem. Products in cohomology.*

Recommended literature: S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton, 1952., E. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966., C. R. F.Maunder, Algebraic Topology, Van Nostrand Reinold, London, 1970., W. S. Massey, Singular Homology Theory, Springer, 1980.

**A representative from the specialized courses**

MDPT3402. Graph theory (Péter Hajnal, CSc)

*Connectivity: the connectivity of directed graphs, nowhere zero flows. Matching: Gallai-Edmonds’ structure theorem, Edmonds’ polytop. random methods for determining ν(G), the number of matchings in a graph, permanent, the proof of Van der Waerden’s conjecture. Coloring of graphs: Hajós’ theorem, Kneser’s graph and its chromatic number, the chromatic number of** **Rd. Independent subsets in graphs: τ-critical graphs, vertex packing polytope, perfect graphs, the Shannon capacity of graphs. Eigenvalues of graphs, random walks on graphs, expanding parameter of graphs.*

Recommended literature: L. Lovász and M.D. Plummer, Matching theory, Akadémiai Kiadó, Budapest, 1986. B. Bollobás, Modern graph theory, Graduate Texts in Mathematics 184., Springer-Verlag, New York, 1998. Reinhard Diestel, Graph theory, Second edition, Graduate Texts in Mathematics 173., Springer-Verlag, 2000.

**5. Stochastics**

**Programme director: Prof. András Krámli DSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 097, fax: (+36 62) 544 548,

E-mail: kramli@informatika.ilab.sztaki.hu

http://www.math.u-szeged.hu/~kramli

**Research topics**

– Asymptotic distributions in probability theory: domains of attraction and partial attraction; St. Petersburg games and merging approximations; weak and strong laws; almost sure asymptotic distributions.

– Statistical estimation and testing: statistical extreme value theory; weighted correlation tests; empirical distributions and transforms; parametric and nonparametric bootstrap; estimation under random censorship and long-range dependence.

– Statistical physics: hyperbolic dynamical systems (hard ball systems); infinite particle systems; mathematical theory of phase transitions.

– Famous problems of probability and statistics from the 17th–19th century and their didactical relations.

Supervisors: A. Krámli DSc, P. Major MHAS, Gy. Pap DSc, K. Stéhlikné Boda PhD, L. Viharos PhD.

**Basic courses**

MDPT251–252. Probability Theory I-II (Gyula Pap, DSc)

*Bernoulli's strong law of large numbers and the DeMoivre-Laplace theorem. Kolmogorov's foundation of probability theory. Random variables, random vectors and their distribution, distribution function. Stochastic processes: Kolmogorov's existence theorem. Independence and product spaces. Discrete, absolutely continuous and singular distributions; Lebesgue's decomposition. Convolutions. Expectation, moments, variance and correlation coefficient. Important special distributions. Types of convergence. Laws of large numbers, 0-1 theorems, three series theorem. Weak convergence, convergence in distribution. Helly's theorem, tightness. Characteristic functions. Central limit theorems. Multivariate normal distributions and multivariate central limit theorems. Local central limit theorems and asymptotic expansions. Conditional probability, conditional expectation and conditional distributions. Random walks. Martingales, Markov chains and stationary sequences. Brownian motion, Gaussian processes: existence and continuity. Nondifferentiablity of the Wiener process. Laws of iterated logarithms, fluctuations. Renewal processes, Poisson process. Combinatorial methods for the random walk, the arc sine law.*

Literature: Billingsley: Probability and Measure, New York, 1986. S. Csörgő: Fifty-three Lectures on Probability, Ann Arbor, 1991 (lecture notes). Feller: Introduction to Probability Theory and its Application I, II, New York, 1968, 1971. Kallenberg: Foundations of Modern Probability, New York, 1997. Petrov: Sums of Independent Random Variables, Berlin, 1975. Spitzer: Principles of Random Walks, New York, 1964.

MDPT253–254. Mathematical statistics I-II (András Krámli, DSc)

*I. Empirical distributions, the Glivenko-Cantelli theorem. Exponential family of distributions. The Fisher information. Theory of point estimates: sufficient statistics, Fisher-Neyman factorization theorem, minimal sufficient statistics, unbiased estimates, consistency, admissibility. The Blackwell-Kolmogorov-Rao theorem, completeness. The Cramér-Rao inequality, efficiency. Estimation methods: the momentum method, the method of minimal divergence, the maximum-likelihood method. The asymptotic properties of the maximum-likelihood method: consistency, asymptotic normality and efficiency. Bayes-estimates: admissibility, unbiasedness, the minimax property. Confidence intervals: exact and asymptotic methods.*

*II. Analysis of contingence tables: the loglinear model. Bias reduction, a short summary of ,,jackknife'' and ,,bootstrap'' methods. The basic notions of statistical hypotheses testing: tests, significance, power of a test, the Neyman-Pearson lemma, unformly most powerful unbiased tests. Monotone likelihood ratios, locally best, invariant, and similar tests. χ2 tests, the classical tests for the parameters of the gaussian distribution, goodness of fit tests. The estimation of the parameters of the multidimensional gaussian distribution. Regression, linear regression, method of lest squares, analysis of variances.*

Literature: Bickel, Doksum: Mathematical Statistics, Oakland 1977. Borovkov: Mathematical Statistics, Amsterdam 1998. Cramér: Mathematical Methods of Statistics, Princeton, 1946. Efron: Bootstrap methods: Another look at the Jackknife, Ann. Statist. 7 (1979) 1-26. Kullback: Information Theory and Statistics, New York 1959. Lehmann: Theory of Point Estimation, New York 1983. Lehmann: Testing Statistical Hypotheses, New York 1983.

MDPT255. Introduction to the ergodic theory (András Krámli, DSc)

*Mean and pointwise ergodic theorems. The theory of maps with discrete spectrum. Examples: the rotation of the unit circle, Bernoulli shift, baker's automorphism, Arnold's CAT. The problems related to the continued fractions.*

Literature: Halmos: Lectures on ergodic Theory, Tokyo, 1956. Kornfeld, Fomin, Sinai: Ergodic Theory, New York, 1982. Khinchin: Kettenbrüche, Lepzig, 1956.

MDPT256. Introduction to the Kolmogorov-Arnold-Moser theory (András Krámli, DSc)

*The n-body problem of classical mechanics, perturbation of Hamiltonian systems, the problem of small divisors. The proof of the twist-lemma and its application to the restricted 3-body problem.*

Literature: Moser: Lectures on Hamiltonian Systems, New York, 1968. Siegel, Moser: Lectures on Celestial Mechanics, New York, 1965

**Two representatives from the specialized courses**

MDPT3508. Foundations of stochastic processes (András Krámli, DSc)

*Discrete** **time and discrete space stochastic processes. Selected topics from the theories of Markov chains, birth-and-death processes, renewal processes, branching processes and discrete time martingals.** *

Literature: Doob: Stochastic Processes, London, 1953. Gikhman, Skorokhod: Introduction to the Theory of Random Processes, Mineola, NY, 1996. Takács: Stochastic Processes London, 1966. Harris: Theory of Branching processes, New York, 1999.

MDPT3506. Extreme-value distributions (László Viharos, PhD)

*The maximum of independent and identically distributed random variables: Gnedenko’s theorem on possible limiting distributions. Domain of attractions of extreme-value distributions. Extension for dependent variables; correlated Gaussian sequences.** *

Literature: de Haan: Regular Variation and Sample Extremes, Amsterdam, 1975. Galambos: The Asymptotic Distribution of Extreme Order Statistics, Malabar, 1987.

**6. Didactics of Mathematics**

**Coordinator:**** ****Assoc. Prof. József Kosztolányi CSc**

Faculty of Science and Informatics, Bolyai Institute

H–6720 Szeged, Aradi vértanúk tere 1., Hungary

Phone: (+36 62) 544 095, fax: (+36 62) 544 548,

E-mail: kosztola@math.u-szeged.hu

This is not a separate educational programme. Each of the five educational programmes above includes topics and courses in the Didactics of Mathematics.

Supervisors: J. Kosztolányi PhD, J. Karsai CSc, L. Klukovits CSc, L. Pintér CSc, I. Szalay CSc, L. Szilassi PhD.

**Major courses*** **(outside the previous five programmes)*

MDPT 261. Selected chapters from the methodology of teaching algebra, number theory, geometry, and combinatorics in high schools and universities (József Kosztolányi, PhD)

MDPT262. Selected chapters from the methodology of teaching analysis, probability theory and statistics in high schools and universities (József Kosztolányi, PhD)

**One representative from the specialized courses**

MDPT3115. Teaching algebra at universities in the twentieth century (Lajos Klukovits CSc)

*The position of algebra at the beginning of the twentieth century. Prerunners and the breakthrough of modern algebra. The comparison of Van der Waerden, Bourbaki, Jacobson, Rédei and Birkhoff’s books. Great inventions of the twentieth century: lattices, categories, universal algebra, homological algebra, etc., and their role in teaching. Algebra in Hungarian education and the corresponding books from G. König till nowadays.** *

Literature: Birkhoff–MacLane: Algebra. Birkhoff–Bartee: Modern Algebra in Computer Science. Corry: Modern Algebra and the Rise of Mathematical Structures. Van der Waerden: Moderne Algebra. Rédei: Algebra. Jacobson: Lectures on Abstract Algebra; General Algebra. Selected chapters from Bourbaki’s books. Selected chapters from Mathematical Intelligencer. Selected chapters from publications on the hystory of mathematics.

**Representative dissertations (title,**** ****author****, supervisor, year)**

- Lattices and invariants,
*E. K. Horváth,*G. Czédli and B. Csákány, 2005 - Asymptotic Bernstein-type inequalities,
*B. Nagy,*V. Totik, 2006 - Bifurcation theory of periodic functional-differential equations,
*G. Röst*, T. Krisztin, 2006 - Smoothness of Green’s function and density of sets,
*F. Toókos*, L. Hatvani and V. Totik, 2006 - The equilibrium measure and the Saff conjecture,
*D. Benkő*, V. Totik, 2006 - Tauber type theorems for ordinary and statistical convergence and statistical limits,
*Á. Fekete*, F. Móricz, 2007 - Absolutely convergent Fourier series and classes of functions spaces,
*V. Fülöp*, F. Móricz, 2007 - Decidability problems in algebra,
*M. Maróti*, Á. Szendrei, 2007 - Minimal clones,
*T. Waldhauser*, B. Csákány and - Approximation of convex bodies by polytopes,
*V. Vígh*, F. Fodor, 2010 Á. Szendrei, 2007 - Orders, conjugacy classes, and coverings of permutation groups,
*A. Maróti*, L. Pyber and Á. Szendrei, 2008 - Orthodox semigroups and semidirect products,
*M. Hartmann*, B. M. Szendrei, 2008 - On stability conditions of operator semigroups,
*Z. Léka*, L. Kérchy, 2009 - Linear combinations of iid random variables from the domain of geometric partial attraction of a semstable law,
*P. Kevei*, S. Csörgő, 2009 - Pointwise convergence of Fourier and conjugate series of periodic functions in two variables,
*Á. Jenei*, F. Móricz, 2009 - Online algoritms for combinatorial problems,
*J. Nagy-György*, P. Hajnal, 2009 - Combinatorical completely 0-simple semigorups and free spectra,
*K. Kátai-Urbán*, L. Megyesi and Cs. Szabó, 2009 - Monoidal intervals,
*M. Dormán*, Á. Szendrei, 2010